Question: If DE bisects ABC, find the value of $x$ if $m\angle ABC = (7x+29)$ degrees and $m\angle DBC= 99$ degrees
1. **State the problem:** We are given that $DE$ bisects $\angle ABC$, and the measures of $\angle ABC$ and $\angle DBC$ are given as $(7x+29)$ degrees and $99$ degrees respectively. We need to find the value of $x$.
2. **Recall the angle bisector property:** If $DE$ bisects $\angle ABC$, then $\angle ABD = \angle DBC$. This means the angle $ABC$ is split into two equal parts.
3. **Set up the equation:** Since $DE$ bisects $\angle ABC$, the measure of $\angle ABC$ is twice the measure of $\angle DBC$.
$$m\angle ABC = 2 \times m\angle DBC$$
Substitute the given values:
$$7x + 29 = 2 \times 99$$
4. **Simplify the equation:**
$$7x + 29 = 198$$
5. **Solve for $x$:**
$$7x = 198 - 29$$
$$7x = 169$$
$$x = \frac{169}{7}$$
6. **Simplify the fraction:**
$$x = 24.142857...$$
So, the value of $x$ is approximately $24.14$ degrees.
**Final answer:**
$$x = \frac{169}{7} \approx 24.14$$